Some results on pseudo MV-algebras with square roots

TL;DR

This paper explores the concept of square roots in pseudo MV-algebras, introducing various notions, classifying them, and characterizing their properties, especially in representable symmetric cases, with concrete examples.

Contribution

It introduces new notions of square roots in pseudo MV-algebras, classifies them, and characterizes their structure in representable symmetric cases using unital ℓ-groups.

Findings

Pseudo MV-algebras with square roots form a proper subvariety.

A strict square root classification is established.

Characterizations of square roots on representable symmetric pseudo MV-algebras are provided.

Abstract

The paper provides a study of pseudo MV-algebras with square roots. We introduce different notions of a square root on a pseudo MV-algebra, and present their main properties. We show that the class of pseudo-MV-algebras with square roots is a proper subvariety of the variety of pseudo MV-algebras. Then, we define a strict square root to classify the class of pseudo MV-algebras with square roots. We found a relationship between strongly atomless pseudo MV-algebras and strict pseudo MV-algebras. Finally, we investigate square roots on representable symmetric pseudo MV-algebras, and we present a complete characterization of a square root and a weak square root on a representable symmetric pseudo MV-algebra using addition in a unital $\ell$-group. Some interesting examples are provided.